Problem 71 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-4 x-8$$

Step-by-Step Solution

Answer

The complete factorization of \(x^{3}+2 x^{2}-4 x-8\) is \((x+2)(x^2+ 4)\) in the real number system

Step 1: Check for any common factors

Start by looking for any factors that are common to all terms in the polynomial. The given polynomial \(x^{3}+2 x^{2}-4 x-8\) does not have any common factors, so we can proceed to the next step.

Step 2: Apply Factoring Techniques

Since there are no common factors, you will need to consider other factoring techniques such as Factoring by grouping or using synthetic/division method. In this case, you can consider synthetic division method to help solve the problem.

Step 3: Start Synthetic Division

One of the roots for the cubic equation can be obtained by using the hit and trial method. Start with \(x = -2\), which confirms that -2 is a root of this polynomial. Then, you use synthetic division to see if the polynomial can be reduced. The synthetic division will look like this:\[\begin{{array}}{cccc}-2 & | & 1 & 2 & -4 & -8 \ & & -2 & 0 & 8 & 0 \ & | & 1 & 0 & 4 & 0\end{{array}}\]The last row (bottom row) tells you that the division is correct and a new simpler polynomial is derived: \(x^2+4\).

Step 4: Factoring further the derived polynomial

We can further factorize \(x^2+4\) as \(x^2 + 2^2\) which is a sum of two squares. But remember, the sum of squares can’t be factorized further in real numbers. Therefore, the complete factorization of \(x^{3}+2 x^{2}-4 x-8\) is \((x+2)(x^2+ 4)\) in the real number system.

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, particularly useful when dividing by a linear expression of the form \(x-a\). This technique is much quicker than the long division method and is especially helpful when finding factors of a polynomial or simplifying polynomial expressions.
To perform synthetic division, choose a suspected root of the polynomial, often derived from simple substitutions. For example, if we want to test if \(x = -2\) is a root of \(x^3 + 2x^2 - 4x - 8\), we place \(-2\) in the synthetic division.

• Write down the coefficients of the polynomial: \(1, 2, -4, -8\).
• Place the root \(-2\) to the left.
• Bring down the leading coefficient \(1\) as it is.
• Multiply the root with the number just brought down and place it under the next coefficient.
• Add these two numbers and write the result beneath.
• Repeat this process until all coefficients have been used.

The result tells us that \(-2\) is indeed a root, and thus the polynomial divides evenly. The simplified polynomial is given by the numbers from the bottom row, excluding the remainder.

Roots of Polynomial

Finding the roots of a polynomial is a crucial step in factoring it fully. A root is a solution to the equation \(f(x) = 0\). In our polynomial \(x^3 + 2x^2 - 4x - 8\), discovering roots can guide the way to factorization.Common methods to find roots include:

• Trial and Error: Plugging in integers to see if the polynomial equals zero.
• Rational Root Theorem: Provides a list of possible rational roots based on the factors of the constant term and the leading coefficient.
• Graphing: Visualizing where the polynomial crosses the x-axis.

In this example, using \(x = -2\) as a root, helps to simplify the polynomial. The availability of real roots can sometimes hint at further factorization or indicate if different techniques like synthetic division might be successful.

Factoring by Grouping

Factoring by grouping is an effective method when you have a polynomial that doesn't have a common factor and is conducive to being split into simpler expressions. It's often useful in polynomials with four terms, just as in our example \(x^3 + 2x^2 - 4x - 8\).Here’s how you generally approach it:

• Group terms that can easily reveal a common factor when considered together. For instance, pair \(x^3 + 2x^2\) and \(-4x - 8\).
• Factor out the greatest common factor from each group. In this example, \(x^2(x + 2)\) and \(-4(x + 2)\).
• Notice if any binomials can be further grouped. If yes, you factor them out, leading to a new expression \((x+2)(x^2-4)\).

While factoring by grouping wasn't the final solution in this case due to the nature of the resulting polynomial, it's an essential technique to carry within your toolkit for other polynomials.

Sum of Squares

The sum of squares form \(a^2 + b^2\) often appears in polynomial equations, but unlike differences of squares, it cannot be easily factored over the real numbers.In our example, the derived polynomial \(x^2 + 4\) takes the form of \(x^2 + 2^2\), which is a sum of squares. This expression can be rewritten but not factored into linear terms in the realm of real numbers.It's important to note that quadratic expressions that appear as a sum of squares do not have real roots. That means their graph does not intersect the x-axis. However, understanding how to recognize and handle such expressions is essential:

• Recognize when polynomials are in the sum of squares form.
• Note that they are not factorable further in the set of real numbers.
• Consider complex numbers if working in a broader number system, where \(a^2 + b^2\) can be rewritten as \((a + bi)(a - bi)\).

Acknowledging these properties helps with determining solutions and understanding the polynomial's behavior.